Question: Factor the following expression: $x^2 - 12x + 35$
Answer: When we factor a polynomial, we are basically reversing this process of multiplying linear expressions together: $ \begin{eqnarray} (x + a)(x + b) &=& xx &+& xb + ax &+& ab \\ \\ &=& x^2 &+& {(a + b)}x &+& {ab} \end{eqnarray} $ $ \begin{eqnarray} \hphantom{(x + a)(x + b) }&\hphantom{=}&\hphantom{ xx }&\hphantom{+}&\hphantom{ (a + b)x }&\hphantom{+}& \\ &=& x^2 & & {-12}x& +& {35} \end{eqnarray} $ The coefficient on the $x$ term is $-12$ and the constant term is $35$ , so to reverse the steps above, we need to find two numbers that add up to $-12$ and multiply to $35$ You can try out different factors of $35$ to see if you can find two that satisfy both conditions. If you're stuck and can't think of any, you can also rewrite the conditions as a system of equations and try solving for $a$ and $b$ $ {a} + {b} = {-12}$ $ {a} \times {b} = {35}$ The two numbers $-5$ and $-7$ satisfy both conditions: $ {-5} + {-7} = {-12} $ $ {-5} \times {-7} = {35} $ So we can factor the expression as: $(x {-5})(x {-7})$